We are having two instances of the website right now. One is the old one containing the cart and a WP instance for pages, posts and gallery functionality. Now, to cut down on development time, we have decided to seamlessly integrate the look & feel of the old website to the new one. And with that, we need to have the following links working from the new website(wordpress side)

The login/logout link. If customer is logged in and would visit one of the WP pages, the section for the My account should reflect the logged in customer's details.

View Cart/ Cart details. The number of checked out items and cost should reflect in the wp website for purchases done within the Shopify domain.

Checkout - same as #2.

For a clearer picture(I hope), the header area looked like this - both in Shopify and our WP theme (can't post images yet):

Home | About | Shop | Gallery

My Account | Logout
View Cart(0 items/$0.0) | Checkout

Any suggestions/inputs to make the bold items working even when in wordpress is greatly appreciated.